Here are the step-by-step solutions for finding the volume of each solid. Remember to round your answers to the nearest tenth as requested.
1) Triangular Prism
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Formula: Volume = Area of Base $\times$ Height ($V = B \cdot h$)
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Base: The base is a triangle with a base of $4\text{ yd}$ and a height of $1.5\text{ yd}$.
* Area of Triangle = $\frac{1}{2} \cdot 4 \cdot 1.5 = 3\text{ sq yd}$.
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Height of Prism: The length connecting the triangular bases is $5\text{ yd}$.
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Calculation: $3 \cdot 5 = 15$.
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Answer: $15.0\text{ yd}^3$
2) Square Pyramid
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Formula: Volume = $\frac{1}{3} \cdot \text{Area of Base} \cdot \text{Height}$ ($V = \frac{1}{3}Bh$)
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Base: A square with side lengths of $3\text{ mi}$ and $4\text{ mi}$? Wait, looking closely at the diagram, the base is a rectangle $3\text{ mi}$ by $4\text{ mi}$ (indicated by the right angle symbol). Let's re-examine. Actually, usually these diagrams imply a square if not specified, but here we have sides labeled 3 and 4 meeting at a corner. So, Base Area = $3 \cdot 4 = 12\text{ sq mi}$.
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Height: The vertical height is given as $5\text{ mi}$.
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Calculation: $V = \frac{1}{3} \cdot 12 \cdot 5$.
* $12 \cdot 5 = 60$.
* $60 / 3 = 20$.
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Answer: $20.0\text{ mi}^3$
3) Square Pyramid
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Formula: $V = \frac{1}{3}Bh$
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Base: A square with side $3\text{ yd}$. Area = $3 \cdot 3 = 9\text{ sq yd}$.
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Height: The vertical height is $5\text{ yd}$.
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Calculation: $V = \frac{1}{3} \cdot 9 \cdot 5$.
* $9 / 3 = 3$.
* $3 \cdot 5 = 15$.
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Answer: $15.0\text{ yd}^3$
4) Cone
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Formula: Volume = $\frac{1}{3} \cdot \pi \cdot r^2 \cdot h$ ($V = \frac{1}{3}\pi r^2 h$)
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Radius ($r$): The diameter is $2\text{ km}$, so the radius is $1\text{ km}$.
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Height ($h$): $3\text{ km}$.
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Calculation: $V = \frac{1}{3} \cdot \pi \cdot (1)^2 \cdot 3$.
* The $\frac{1}{3}$ and the $3$ cancel out.
* $V = \pi \cdot 1 \approx 3.14159...$
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Rounding: Round to the nearest tenth: $3.1$.
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Answer: $3.1\text{ km}^3$
5) Cylinder
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Formula: Volume = $\pi \cdot r^2 \cdot h$ ($V = \pi r^2 h$)
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Radius ($r$): The diameter is $4\text{ in}$, so the radius is $2\text{ in}$.
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Height ($h$): The length of the cylinder is $3\text{ in}$.
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Calculation: $V = \pi \cdot (2)^2 \cdot 3$.
* $2^2 = 4$.
* $4 \cdot 3 = 12$.
* $V = 12\pi \approx 37.699...$
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Rounding: Round to the nearest tenth: $37.7$.
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Answer: $37.7\text{ in}^3$
6) Cube
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Formula: Volume = side $\cdot$ side $\cdot$ side ($V = s^3$)
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Side ($s$): $2\text{ m}$.
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Calculation: $2 \cdot 2 \cdot 2 = 8$.
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Answer: $8.0\text{ m}^3$
7) Triangular Prism
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Formula: Volume = Area of Base $\times$ Height ($V = B \cdot h$)
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Base: The base is the triangle on the front.
* Base of triangle = $6\text{ yd}$.
* Height of triangle = $2.5\text{ yd}$.
* Area of Triangle = $\frac{1}{2} \cdot 6 \cdot 2.5 = 3 \cdot 2.5 = 7.5\text{ sq yd}$.
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Height of Prism: The length connecting the triangles is $3\text{ yd}$.
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Calculation: $7.5 \cdot 3 = 22.5$.
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Answer: $22.5\text{ yd}^3$
8) Rectangular Pyramid
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Formula: Volume = $\frac{1}{3} \cdot \text{Area of Base} \cdot \text{Height}$ ($V = \frac{1}{3}Bh$)
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Base: A rectangle with dimensions $2\text{ in}$ and $1\text{ in}$.
* Area of Base = $2 \cdot 1 = 2\text{ sq in}$.
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Height: The vertical height is $1\text{ in}$.
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Calculation: $V = \frac{1}{3} \cdot 2 \cdot 1$.
* $V = \frac{2}{3} \approx 0.666...$
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Rounding: Round to the nearest tenth: $0.7$.
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Answer: $0.7\text{ in}^3$
Final Answer:
1) 15.0 yd³
2) 20.0 mi³
3) 15.0 yd³
4) 3.1 km³
5) 37.7 in³
6) 8.0 m³
7) 22.5 yd³
8) 0.7 in³
Parent Tip: Review the logic above to help your child master the concept of volumes of solids worksheet.
